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So I know all the elements of $S_4=\{e,(12),(13),(14),(23),(24),(34),(12)(34),(13)(24),(12)(23),(123),(132),(124),(142),(134),(143),(234),(243),(1234),(1342),(1423),(1243),(1432),(1324)\}$

I also understand that $e, (12), (12)(34), (123)$, and $(1234)$ are the classes. What's confusing me is how to find the number of elements in each class. For example I see ($12)(12)(12)^{-1}=(12)$ but when we look at the next element we see $(13)(12)(13)^{-1}=((13)(1) (13)(2))$.

How do we simplfy this/ how are we using number theory here to get elements of these classes?

Sebastiano
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  • Relevant: https://math.stackexchange.com/questions/274657/bijection-between-permissible-cycle-types-and-conjugacy-classes – Theo C. Oct 02 '19 at 21:19
  • Answer from J.Schmidt in this post may help you : https://math.stackexchange.com/a/379912/399263 – zwim Oct 02 '19 at 21:26
  • Did you learn about cycle types? Each cycle type corresponds to a conjugacy class. This should make the rest of the exercise easy. – Ayman Hourieh Oct 02 '19 at 21:29
  • Welcome to Mathematics Stack Exchange. Note that $(12)(23)$ in your list of elements is not in disjoint cycle form like the others you listed; perhaps you meant $(14)(23)$ – J. W. Tanner Oct 02 '19 at 21:38

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